What are the solutions to (x-2)(x+4) = 0?

Rationale

-4 and 2

To solve the equation (x-2)(x+4) = 0, we can apply the zero product property, which states that if the product of two factors is zero, at least one of the factors must be zero. Setting each factor equal to zero gives us the solutions x - 2 = 0 and x + 4 = 0, leading to x = 2 and x = -4.

A (-4 and 2) CORRECT

This choice correctly identifies the solutions to the equation. Setting each factor to zero results in the values x = 2 and x = -4, confirming that these are indeed the correct solutions for the given equation.

B (-3 and 1) YOUR ANSWER

This option suggests incorrect solutions that do not satisfy either of the factors in the original equation. Plugging in -3 or 1 into (x-2)(x+4) does not yield a product of zero, indicating that neither value is a solution.

C (-2 and 4) YOUR ANSWER

Similar to option B, these values do not satisfy the original equation. Testing -2 or 4 in (x-2)(x+4) does not yield zero, thus disqualifying them as solutions to the equation.

D (-1 and 1) YOUR ANSWER

This choice presents another incorrect pair of values. Substituting -1 or 1 into the factors of the equation also fails to produce a product of zero, confirming they do not solve the equation.

E (-1 and 3) YOUR ANSWER

This option, like the previous ones, does not provide valid solutions. Neither -1 nor 3 results in zero when substituted into the factors of the equation, making them incorrect.

Conclusion

The solutions to the equation (x-2)(x+4) = 0 are indeed -4 and 2, as derived from setting each factor to zero. All other options present values that do not satisfy the equation, underscoring the importance of applying the zero product property correctly in identifying solutions.